65. this question has two parts. first, answer part a. then, answer part b. in the figure below, $\triangle fgh \cong \triangle jkl$. part a: what is the value of $x$? part b: what is the value of $y$? 66. this question has two parts. first, answer part a. then, answer part b. in the figure below, $\angle a \cong \angle d$, $\angle b \cong \angle e$, and $\overline{ab} \cong \overline{de}$. part a: are the triangles congruent? if so, what congruence theorem proves they are congruent? a. yes, the triangles are congruent by the side - angle - side congruence theorem. b. yes, the triangles are congruent by the angle - angle - side congruence theorem. c. yes, the triangles are congruent by the angle - side - angle congruence theorem. d. no, the triangles are not congruent. part b: if the triangles are congruent, find the length of $\overline{df}$. e. 43 f. 6 g. 40 h. cannot be determined because the triangles are not congruent. 67. which congruence statement is correct? a. $\triangle abc \cong \triangle mnp$ b. $\triangle acb \cong \triangle mpn$ c. $\triangle cab \cong \triangle nmp$ d. $\triangle bca \cong \triangle pmn$ 68. the triangles in the diagram below are congruent. if $\triangle jkl \cong \triangle onm$ then what is $m\angle n$?

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Accepted Answer

### Question 65 Part A
# Explanation:
## Step1: Find angle at J
In $	riangle FGH$, sum of angles is $180^\circ$. So $\angle F = 180 - 138 - 20 = 22^\circ$. Since $	riangle FGH \cong 	riangle JKL$, $\angle J = \angle F = 22^\circ$.
## Step2: Solve for x
Set $3x - 8 = 22$. Then $3x = 30$, so $x = 10$.
# Answer:
$x = 10$

### Question 65 Part B
# Explanation:
## Step1: Find side length
Since $	riangle FGH \cong 	riangle JKL$, $LH = FJ$. $LH = 14.7$ in, so $\frac{1}{4}x + y = 14.7$.
## Step2: Substitute x and solve for y
We know $x = 10$, so $\frac{1}{4}(10) + y = 14.7$. $2.5 + y = 14.7$, so $y = 12.2$.
# Answer:
$y = 12.2$

### Question 66 Part A
# Explanation:
We have $\angle A \cong \angle D$, $\angle B \cong \angle E$, and $\overline{AB} \cong \overline{DE}$. By Angle - Side - Angle (ASA) congruence theorem (two angles and the included side), the triangles are congruent. So option C is correct.
# Answer:
C. Yes, the triangles are congruent by the Angle - Side - Angle Congruence Theorem.

### Question 66 Part B
# Explanation:
## Step1: Find x
Since triangles are congruent, $6x + 4 = 4x + 19$ (corresponding sides). $2x = 15$, $x = 7.5$. Then $DF = 4x + 19 = 4(7.5)+19 = 30 + 19 = 49$? Wait, no, wait. Wait, $\overline{AC}=7x + 1$, $\overline{AB}=6x + 4$, $\overline{DE}=4x + 19$. Wait, maybe I made a mistake. Wait, if ASA, then $AC = DF$? Wait, no, let's re - check. Wait, $\angle A \cong \angle D$, $\angle B \cong \angle E$, $\overline{AB} \cong \overline{DE}$. So $	riangle ABC \cong 	riangle DEF$ (ASA). So $AC = DF$, $AB = DE$, $BC = EF$. Let's find $x$ from $AB = DE$: $6x + 4 = 4x + 19$ gives $x = 7.5$. Then $AC = 7x + 1 = 7(7.5)+1 = 52.5 + 1 = 53.5$, $DF$ should be equal to $AC$? Wait, no, maybe the sides: $AB = 6x + 4 = 6(7.5)+4 = 45 + 4 = 49$, $DE = 4x + 19 = 4(7.5)+19 = 30 + 19 = 49$. Then $AC = 7x + 1 = 53.5$, $DF$ should be equal to $AC$? Wait, no, maybe the problem is that $DF$ is corresponding to $AC$? Wait, maybe I messed up. Wait, the options are E.43, F.6, G.40, H. Cannot be determined. Wait, maybe I made a mistake in the side correspondence. Let's re - do. If $	riangle ABC \cong 	riangle DEF$ (ASA), then $AB = DE$, $BC = EF$, $AC = DF$. So $AB = 6x + 4$, $DE = 4x + 19$. So $6x + 4 = 4x + 19$ gives $x = 7.5$. Then $AC = 7x + 1 = 7(7.5)+1 = 53.5$, but that's not one of the options. Wait, maybe the sides are $AB = 6x + 4$, $AC = 7x + 1$, and in the other triangle $DE = 4x + 19$, $DF =?$ Wait, maybe the problem is that $DF$ is equal to $AB$? No, that doesn't make sense. Wait, maybe the correct way is: since $	riangle ABC \cong 	riangle DEF$, then $DF = AC$. Wait, $AC = 7x + 1$, $x = 7.5$, $AC = 7*7.5 + 1 = 52.5 + 1 = 53.5$, which is not an option. Wait, maybe I made a mistake in the congruence. Wait, the problem says "find the length of $\overline{DF}$". Wait, maybe the sides are $AB = 6x + 4$, $BC =?$, $AC = 7x + 1$ and in $	riangle DEF$, $DE = 4x + 19$, $EF =?$, $DF =?$. Wait, maybe the correct equation is $6x + 4 = 4x + 19$ (AB = DE), $x = 7.5$. Then $DF = 6x + 4$? No, that's AB. Wait, maybe the answer is E.43? Wait, if $x = 6$, then $6x + 4 = 40$, $4x + 19 = 43$. Wait, maybe I made a mistake in the equation. Wait, maybe $7x + 1 = 4x + 19$? Then $3x = 18$, $x = 6$. Then $DF = 4x + 19 = 4*6 + 19 = 24 + 19 = 43$. Ah, that's option E. So I must have messed up the side correspondence. So the correct equation is $AC = DF$, so $7x + 1 = 4x + 19$, $3x = 18$, $x = 6$. Then $DF = 4*6 + 19 = 43$.
# Answer:
E. 43

### Question 67
# Explanation:
We need to match the congruent triangles by their corresponding angles and sides. Looking at the markings: $\angle A$ and $\angle N$ have the same marking, $\angle B$ and $\angle M$ have the same marking, $\angle C$ and $\angle P$ have the same marking. And the sides: $AC$ and $NP$ (same marking), $AB$ and $NM$ (same marking), $BC$ and $MP$ (same marking). So $	riangle CAB \cong 	riangle NMP$ (corresponding angles and sides). So option C is correct.
# Answer:
C. $	riangle CAB \cong 	riangle NMP$

### Question 68
# Explanation:
## Step1: Find angle in $	riangle JKL$
In $	riangle JKL$, $\angle J = 60^\circ$, $\angle K =?$, $\angle L =?$. Since $	riangle JKL \cong 	riangle ONM$, $\angle N = \angle K$. In $	riangle ONM$, $\angle M = 56^\circ$, $\angle O =?$, $\angle N =?$. The sum of angles in a triangle is $180^\circ$. In $	riangle JKL$, $\angle J = 60^\circ$, so $\angle K + \angle L = 120^\circ$. Since $	riangle JKL \cong 	riangle ONM$, $\angle N = \angle K$. In $	riangle ONM$, $\angle M = 56^\circ$, so $\angle N = 180 - 60 - 56 = 64^\circ$? Wait, no. Wait, $	riangle JKL \cong 	riangle ONM$, so $\angle J \cong \angle O$, $\angle K \cong \angle N$, $\angle L \cong \angle M$. $\angle M = 56^\circ$, so $\angle L = 56^\circ$. In $	riangle JKL$, $\angle J = 60^\circ$, $\angle L = 56^\circ$, so $\angle K = 180 - 60 - 56 = 64^\circ$. So $\angle N = \angle K = 64^\circ$. Wait, but the diagram shows $\angle M = 56^\circ$, $\angle J = 60^\circ$. So $\angle N = 180 - 60 - 56 = 64^\circ$.
# Answer:
$m\angle N = 64^\circ$

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QUESTION IMAGE

Question

Explanation:

Question 65 Part A

Step1: Find angle at J

Step2: Solve for x

Step1: Find side length

Step2: Substitute x and solve for y

Answer:

Question 65 Part B